Maximum Mean Discrepancy

Federated learning trains a shared model from private client data. In practice, data-generating distributions may differ, and the true mixture across clients is often unknown, making the underlying group distribution difficult to specify. Existing approaches address cross-client mixture uncertainty by optimizing against the worst-case mixture, yet assume accurate client-wise distribution estimates. However, these estimates can be unreliable when based on finite samples. To handle both cross-client mixture uncertainty and within-client distributional ambiguity, we construct a global ambiguity set as the union of admissible mixtures of local ambiguity sets. The construction allows client-specific ambiguity radii and admits a client-wise separable reformulation. Leveraging this structure, we establish a high-probability out-of-sample performance guarantee. We further develop a federated algorithm for a penalty-based reformulation and prove its convergence under milder regularity conditions. Simulations validate the algorithm's effectiveness.

We study federated soft clustering over federated learning (FL) networks of devices that each hold a private local dataset and fit a personalized Gaussian mixture model (GMM). Generalized total variation minimization (GTVMin) couples the local maximum likelihood problems through a graph regularizer that penalizes a discrepancy between the models of connected nodes. The choice of discrepancy measure is a key design decision: we compare a squared Euclidean distance between model parameters, which requires component matching, with two measures that compare the local model distributions directly and hence need no matching: a Monte-Carlo approximated Kullback-Leibler (KL) divergence and a closed-form maximum mean discrepancy (MMD). All three resulting GTVMin instances are optimized by synchronous projected gradient updates; for the smooth MMD instance we provide a convergence guarantee to stationary points. We characterize their computational cost and evaluate their robustness to data heterogeneity.

Multiple kernel $k$-means integrates complementary nonlinear similarities by learning a combination of base kernels. Its pointwise optimization, however, is sensitive to noisy and boundary samples and repeatedly operates on sample-scale kernel matrices. Granular-ball representations organize local sample groups into mesoscopic units, but granular balls generated once in the input space may be inconsistent with the fused-kernel geometry that evolves during multiple kernel learning. We propose dynamic kernel-space granular-ball multiple kernel $k$-means (DK-GBMKKM). The method generates granular balls in the current fused kernel space and alternates kernel-weight learning with granular-ball membership updates, allowing the representation to adapt to changes in the fused-kernel geometry. A sample-size-weighted granular-ball kernel is further constructed to preserve the contributions of balls of different sizes, and its positive semidefiniteness and related equivalence properties are established. Experiments on 12 public datasets demonstrate the strong overall clustering performance of DK-GBMKKM. The code has been open-sourced for reproducibility: https://github.com/lianxiaoyu724/DK-GBMKKM.

Benchmark leaderboards summarize how well a language model performs, but not how its behavior relates to that of other models or changes across generations. We characterize the output behavior of 32 models from six families using their responses to a shared bank of 10{,}000 prompts. After embedding each response, we construct three complementary sentence-level dissimilarities: an aligned mean per-prompt distance, which is a pseudometric on observed model responses; a PCA-compressed summary of prompt-wise disagreement; and an alignment-free Gromov--Wasserstein discrepancy between models' internal response geometries. We use these constructions to study static organization and temporal change on a release-date axis through behavioral maps, family-wise drift, hierarchical clustering, cross-family convergence, and response-cloud dispersion. Across the three constructions, model families form coherent clusters, with \texttt{gpt-2} as a global outlier; cross-family distances decrease over time; and several recent reasoning-oriented models have comparatively compact response clouds. A token-level cross-check based on per-prompt Maximum Mean Discrepancy closely agrees with the sentence-level mean distance (Spearman $ρ=0.98$) and recovers the same qualitative findings. We organize these comparisons through a measure-theoretic lens making their alignment and invariance assumptions explicit. We also establish an architecture-agnostic sufficient condition linking behavioral similarity to inference-prompt coverage, small excess population log-loss, and similar effective target distributions---a possible training-side account rather than an empirical explanation of the observed trends. Our pipeline is label-free, and re-encoding every response with three further encoders---down to one $73\times$ smaller---preserves the rank geometry, the outliers, and the sign of the time trend.

Fair representation learning with a continuous sensitive attribute $S$ requires a representation $Z$ that is statistically independent of $S$. Existing criteria, including generalized demographic parity, the expectation of integral probability metrics (EIPM), and mutual information, enforce this independence by averaging a per-value discrepancy between the conditional law $P_{Z \mid S=s}$ and the marginal $P_Z$ over the law of $S$. This approach requires a nonparametric surrogate for the conditional law at each sensitive value. We propose evaluating independence through a single joint discrepancy $d\left(P_{Z, S}, P_Z \otimes P_S\right)$ between the joint law and the product of its marginals. We establish a disintegration identity; on decomposable witness classes it equals the conditional-integral functional that EIPM and generalized demographic parity instantiate. By reaching the same target without the conditional law, this discrepancy can be estimated directly from samples via a dependence statistic rather than conditional smoothing. We take the Hilbert-Schmidt independence criterion (HSIC) as an instance of the joint discrepancy $d$ to investigate the statistical efficiency of replacing the conditional formulation. The HSIC estimator is a closed-form $O\left(n^2\right)$ statistic that converges at the $O\left(n^{-1 / 2}\right)$ rate, in contrast to the nonparametric $O\left(n^{-2 / 5}\right)$ rate of the conditional-route estimators. We prove this instance is equivalent to the conditional maximum mean discrepancy (MMD) integral up to an explicit spectral tail. The corresponding algorithmic implementation, i.e., FRHSIC, attains fairness-accuracy tradeoffs comparable to conditional-route basel es while reducing per-epoch training time.

Kernel spectral clustering with a single bandwidth can be inadequate for data exhibiting multiple characteristic pairwise-distance scales, a problem particularly prevalent in the high-dimensional regime. We address this issue through a multi-kernel formulation that aggregates kernels with different bandwidths. The bandwidths are selected as prescribed empirical quantiles of the pairwise squared distances, thereby capturing the relevant distance scales without requiring prior population-scale information. We develop a rigorous theoretical analysis of the resulting method under a general high-dimensional, multi-scale mixture model with heterogeneous cluster centers and covariance geometries. We construct a blockwise constant, low-rank informative approximation to the empirical multi-kernel matrix and establish row-wise $\ell_{2,\infty}$ perturbation bounds for its leading spectral components, as well as for the associated normalized Laplacian matrix. These bounds yield observation-level control of the spectral embedding, which is more informative than conventional global eigenspace perturbation estimates. Under suitable eigen-gap and cluster-separation conditions, we show that approximate $K$-means applied to the multi-kernel spectral embedding achieves exact recovery with high probability.

Detecting that a stream of high-dimensional embeddings has changed is usually framed as a choice of statistic. We give a scale law that constrains any moment-based choice and test it against topological alternatives. The law: certifying a feature of spatial scale eps carrying mass fraction f requires polynomial tests of degree N* >= log(1/f)/(2 eps), proved via the Chebyshev extremal problem; a Gauss-quadrature construction gives N* >= 4b-1 for a b-scale topology, so cost is set by feature fineness, not feature count. The law is one-sided: we exhibit an annulus whose mean, covariance and all fourth-order moments equal those of a filled disk, yet H_1 is nonzero. Its practical content is a calibration rule. The upper bound is attained by Gaussian test functions, the RKHS witness of an RBF kernel, so the law predicts which bandwidth an MMD test should use: the feature scale. On real embedding streams we measure sigma*/eps with median 1.12 (IQR 1.01-1.52, n=26) over three settings and three scales, and a data-driven bandwidth reaches AUC >= 0.95. Against an adversary optimised against the defender's statistics (mean, covariance, k-NN, kurtosis), only a bandwidth-matched kernel test still detects. For persistent homology the verdict is mixed and depends on choices usually left implicit. The summary matters more than the filtration: total persistence attains recall 0.75 at FPR 1% where the first persistence landscape attains 0.00. What survives is a cost gap, not a power gap: where persistence works it costs 116x kurtosis, which works at least as well. We conclude not that topological summaries are useless, but that on this task a kernel test whose bandwidth the law sets dominates them.

Accurate model evaluation in machine learning depends critically on how datasets are split into training and testing subsets. Standard random splitting assumes that both partitions share the same underlying distribution, an assumption often violated in datasets with class imbalance, natural clustering, or spatial autocorrelation. This paper investigates the role of statistical similarity in train-test splitting and its consequences for AutoML model evaluation. Five established strategies are compared across fifteen UCI benchmark datasets: random splitting, stratified sampling, Kennard-Stone, Duplex, and SPXY. Similarity is assessed using chi-square, Kolmogorov-Smirnov, and Maximum Mean Discrepancy (MMD) tests. Geometry-based methods consistently produce near-zero MMD scores, introducing instability into downstream performance estimates. The proposed Optimised-Distribution method treats similarity as an explicit optimisation objective and achieves the highest mean MMD similarity, 89.0%, across all strategies evaluated.

Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others. Their fastest estimators are known to converge at a parametric rate---$n^{-1/2}$---under mild conditions. While this rate is known to be minimax optimal on $\mathbb R^d$ under strict assumptions with bounded kernels, little is known about its optimality beyond the finite-dimensional Euclidean setting with unbounded kernels. In this work, we prove that the minimax lower bound of estimation of the most popular kernel discrepancies (maximum mean discrepancy, Hilbert-Schmidt independence criterion and kernel Stein discrepancy; MMD, HSIC, KSD) is $n^{-1/2}$ on general topological spaces, and under mild assumptions on the kernel; the same rates are shown (as corollaries) to hold for the estimation of the mean embedding and the centered cross-covariance operator. Our results settle the question of optimal estimation of these kernel discrepancies.

This paper studies the problem of stochastic variance reduction (SVR) for the maximum mean discrepancy (MMD) and correlation alignment (CORAL) loss functions. Although various offline SVR algorithms for these losses have been proposed, these are incompatible with online, distributed, or incremental learning settings. This paper presents Adaptive vaRiance Reduction via Online reWeighting (ARROW), the first online SVR algorithm for the MMD and CORAL for streamed data. The method maintains moving average references of the alignment statistics, and adaptively reweights incoming minibatches so that the minibatch and reference statistics are aligned. Further, we propose a relaxed reweighting scheme so that the ensuing weight-optimisation problem is tractable. In experiments and simulations, we show that ARROW performs competitively with offline algorithms in terms of runtime, degree of variance reduction achieved, and target domain accuracy.

We introduce the Directional Kernel Mean Difference (DKMD), a signed statistic for univariate distribution comparison that preserves the direction of distributional shifts. Unlike the squared Maximum Mean Discrepancy (MMD), which discards directional information by squaring the RKHS distance, DKMD integrates the difference of kernel mean embeddings against a fixed odd weighting function. This construction yields three structural properties: antisymmetry, immunity to symmetric distributional differences, and directional monotonicity under stochastic dominance. We derive a data-driven Riemann estimator that ensures asymptotic consistency with the continuous formulation, strictly preserving the theoretical guarantees of the signed statistic in empirical evaluations. To overcome the quadratic computational cost of kernel methods, we develop an $O(N \log N)$ prefix--suffix scanning algorithm that exploits the total order of the real line while requiring only $O(N)$ memory. Experiments on synthetic benchmarks demonstrate that DKMD correctly isolates directional shifts from symmetric perturbations, remains robust to heavy-tailed outliers that can flip the sign of the mean difference, and scales to millions of samples in seconds.

Big data clustering remains challenging: the Minimum Sum-of-Squares Clustering (MSSC) problem underlying K-means is NP-hard, and existing methods either reach poor local minima or require prohibitive metaheuristic hybrids. We target arbitrarily tall data: a fixed feature space may contain arbitrarily many, possibly infinitely many, observations, while the algorithm accesses only finite random samples. We propose Big-means++, an algorithm achieving scalability and global-search quality by curating inputs to MSSC optimization on big data. It orchestrates local K-means refinements into a data-native global search for big data clustering. Rather than optimizing the full-data MSSC objective, Big-means++ traverses sample-induced surrogate landscapes. Each sample defines a distinct empirical MSSC approximation with a perturbed local-optimum structure, turning sample-to-sample variation into a global-search mechanism. Unlike Big-means, a flowing-incumbent strategy propagates centroid state across empirical landscapes through K-means refinements on fresh samples without rollback to a best-so-far solution. This increases mobility and favors stable, high-quality configurations across approximations of the full-data structure. A new shaking mechanism varies sample size geometrically, broadening the surrogate landscapes explored across resolution scales, accounting for cluster imbalance, and improving solution quality. A competitive multi-agent system asynchronously explores independent sampled landscapes, transforming diverse stochastic trajectories into collective search intelligence. Automatic convergence detection stops each agent after attaining a high-quality solution but before further search risks degrading it, while providing a universal speed-quality control. Experiments on 22 datasets against 11 competing algorithms demonstrate the effectiveness, efficiency, and robustness of Big-means++.

Safe reinforcement learning typically enforces safety by bounding expected cumulative costs, a criterion that often fails to detect rare but catastrophic tail events. To overcome these limitations, this paper introduces SteinGate, a boundary-aware distributional safety certificate that replaces fragile tail fitting with a robust consistency check using Kernelized Stein Discrepancy while accounting for boundary atoms induced by clipped costs. SteinGate evaluates whether observed policy rollout costs remain consistent with a safe reference distribution, providing a non-parametric safety certificate. This certificate is used to dynamically adapt the learning regime: favoring reward-improving policy updates when rollouts remain consistent with the safe reference and switching to recovery behavior when the cost tail deviates. Experiments on continuous-control benchmarks demonstrate that SteinGate significantly reduces both the frequency and severity of constraint violations during training while maintaining competitive returns relative to state-of-the-art baselines.

The k-means++ algorithm is commonly restarted multiple times to avoid poor local optima, yet the number of restarts is almost always chosen arbitrarily and applied uniformly regardless of data set difficulty. This undermines any comparison relying on such a choice and wastes computation on easy data sets while potentially under-serving hard ones. We introduce GTRC, a restart criterion combining a Good-Turing estimate, a proven unconditional bound, and a confidence-based bound on the probability that a further restart would improve on the current result, stopping once this probability falls below a user-specified tolerance $\varepsilon$. Across 36 data sets, GTRC reached clustering quality competitive with well-chosen fixed restart counts, while the number of restarts used varied considerably and appropriately with data set difficulty, governed by an interpretable, data-dependent signal rather than a fixed rule. GTRC offers a principled and reportable alternative to fixing the number of $k$-means++ restarts in advance. Software:https://github.com/RCdeAmorim/Good-Turing-Restart-Criterion.

We present the dithered Gaussian mechanism, a novel alternative to the discrete Gaussian mechanism for differential privacy that discretizes the private output rather than the noise distribution itself. By interpreting this discretization as post-processing of the Gaussian mechanism, our construction directly inherits the privacy guarantees of the standard Gaussian mechanism while avoiding vulnerabilities caused by finite-precision floating-point outputs. We show that the mechanism is provably randomness-efficient: by sampling the discretized output values directly, the number of high-quality random bits required for privacy can be reduced significantly and made independent of the noise level. This is achieved by separating the randomness into two sources: a high-quality source used for the privacy-critical sampling step, and a high-performance public source, possibly known to the adversary, that supplies the additional randomness needed for randomized discretization. This separation enables the use of cryptographically secure randomness without substantial performance loss. As an application, we study model training with DP-SGD and show that cryptographically secure noise generation with reduced exposure to floating-point vulnerabilities can be achieved with modest practical overhead.

Conditional generative models have emerged as powerful tools for sampling from target conditional distributions, driving substantial advances across a wide range of scientific and applied domains. As these models proliferate, practitioners often face multiple plausible generators whose performance can vary with the task, data, or input condition. We propose an optimal model averaging framework for conditional generative models, allowing candidate generators to be combined even when they are accessible only through conditional samples without tractable densities. Specifically, we use a sample-based maximum mean discrepancy between conditional distributions, which first leads to a static model averaging method, StaticMA, assigning fixed weights to different candidates. In addition, we develop MoEMA (mixture-of-experts model averaging), an input-adaptive method that parameterizes covariate-dependent weights through a softmax neural-network gate. We establish in-sample and out-of-sample asymptotic optimality for the proposed methods, together with consistency of the estimated adaptive weight function under regularity conditions. The framework applies directly to Euclidean responses and extends to unstructured data by combining our formulation with fixed representation maps. Across a broad set of simulations and real-data studies spanning tabular, image, and text modalities, MoEMA generally improves over competing baselines, demonstrating the effectiveness of our proposed methods.

Minimum maximum mean discrepancy (MMD) estimation has emerged as a robust and likelihood-free alternative to maximum likelihood estimation for parameter estimation. Yet, despite its practical success, the associated optimization problem remains poorly understood, with theoretical guarantees for existing algorithms hinging on convexity assumptions that rarely hold in practice. We address this gap by proposing a preconditioned gradient descent (PGD) scheme, establishing its asymptotic \emph{global} convergence under explicit gradient-dominance and projection-residual conditions. Our approach is inspired by recent progress on MMD gradient flows, a nonparametric descent scheme on the space of probability measures. We provide extensive empirical evidence that our PGD scheme outperforms standard gradient descent across a range of challenging parameter estimation and composite hypothesis testing problems.

The classical $k$-means clustering cannot be directly used to incomplete data, and existing $k$-means-based clustering for missing data primarily focus on improving the practical accuracy of clustering, whereas most of them lack theoretical guarantees in the asymptotic sense. In this paper, we investigate the statistical properties of $k$-means clustering in the presence of missing data. We first establish the $\sqrt{n}$-excess risk bound and prove the consistency of the estimated cluster centers under general missing mechanisms. For the Missing Completely at Random (MCAR) mechanism, we further derive the $\sqrt{n}$-convergence rate and asymptotic normality of the estimated cluster centers. Moreover, we study in what cases the cluster centers estimated by incomplete data converge to the true cluster centers of original fully observed data, and give a sufficient condition about the missing probability and the separation among true clusters. These results provide a theoretical guarantee for missing-data-$k$-means. Notably, our analysis reveal that under MCAR mechanism, both achieving the $\sqrt{n}$-rate and converging to the true cluster centers require $k$ true centers to be distinct in every dimension, highlighting the significant challenges of application in high-dimensional regimes. Finally, we conduct numerical simulations on synthetic incomplete datasets to support our theoretical analysis results.

We present MMD-Reg, a novel correspondence-free approach to point-cloud registration that is differentiable and has linear computational complexity in the number of points. We model registration as a nonlinear least-squares problem based on the Maximum Mean Discrepancy, approximated using random Fourier features. The resulting objective can be solved efficiently with standard methods such as Levenberg-Marquardt, and the solution is differentiable via the implicit function theorem. This allows MMD-Reg to be used as a differentiable optimization layer within end-to-end trainable models, supporting registration under challenging conditions such as poor initial alignment and partial overlap. We demonstrate this Neural MMD-Reg formulation by integrating the layer with a set transformer, training the resulting model in supervised and unsupervised settings, and comparing its performance against recent learning-based methods. We also evaluate standalone MMD-Reg, comparing its accuracy and scalability against widely used non-learning-based registration methods.

Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.

Despite their strong general capabilities, large language models (LLMs) often remain unreliable when outputs must be numerically precise. A key reason is the training objective: standard cross-entropy treats numeric tokens as unstructured categories and ignores the metric structure of their values. We address this mismatch with Smooth Maximum Mean Discrepancy (SMMD), which builds on the classic MMD by incorporating value-distance kernels over numeric tokens and graph-based smoothness. With this kernel defined over a numeric sub-vocabulary, SMMD aligns the predicted numeric distribution to the target via kernel matching and smooths the prediction-target residual over the induced kernel graph to encourage local consistency. We evaluate SMMD on four numeric-target tasks: mathematical reasoning, arithmetic calculation, clock-time recognition, and chart question answering, across multiple open-weight LLM and VLM backbones. SMMD consistently improves accuracy over both cross-entropy and recent numeric-target losses; analyses show complementary effects between MMD and smoothness and underscore the importance of distance-based kernel design. Code is available at https://github.com/Zuozhuo/smmd-loss.

Generative models can produce individually plausible samples while deviating substantially from a target set in the distribution of key features. For example, a model pretrained on broad drug-like chemical space may generate molecules whose molecular features differ from those of a therapeutic class of interest, such as known antibiotics. Correcting such distributional miscalibration is challenging: direct finetuning on the target set can overfit and does not control which features are matched. To fill this gap, we introduce kernel Calibrating Generative Models (kCGM). kCGM minimizes a maximum mean discrepancy (MMD) between generated and target feature distributions using an unbiased score-function estimator, with KL regularization to remain close to the pretrained model. On a target set of 174 antibiotics, direct finetuning sacrifices chemical validity for feature-distribution matching, whereas kCGM improves target feature matching while increasing validity. We further demonstrate kCGM in protein and DNA generation tasks, showing it can adapt autoregressive, continuous-space diffusion, and discrete diffusion models using only feature-level supervision. Code is available at https://github.com/smithhenryd/cgm.

In Self-Supervised Learning (SSL), preventing representation collapse by explicitly enforcing a uniform distribution on the unit hypersphere has proven to be effective. However, current frameworks typically rely on sliced statistical regularizers such as SIGReg (used in LeJEPA) and SUSReg (used in SPHERE-JEPA), which approximate this continuous objective via Monte Carlo sampling along random 1D directions. This stochasticity injects projection variance into the training gradients, destabilizing optimization, and hindering convergence. In this work, we first show that analytically integrating out these random projections natively yields a deterministic Maximum Mean Discrepancy (MMD), bypassing the variance of sliced methods. Motivated by this equivalence, we formulate full-dimensional objectives for MMD, Kernel Stein Discrepancy (KSD), and Kullback-Leibler (KL) divergence directly on the sphere to enforce a uniform distribution. To prevent spatial bias, we equip these tests with rotationally invariant kernels constructed via spectral theory, systematically evaluating two canonical families: smooth exponential decay (Heat) and strict frequency cutoff (Bandlimited) filters. Empirically, removing projection-induced noise results in more stable optimization, faster convergence, and consistent improvements over stochastic sliced regularizers on ImageNet and Galaxy10. Furthermore, we reveal that the choice of the statistical test shapes the geometry of the learned latent space: MMD and KSD favor locally clustered organization suitable for object-centric domains, whereas the continuous KDE-based KL divergence promotes fine-grained instance separation, yielding the strongest results on unclustered procedural texture retrieval.

Vision-language foundation models such as CLIP and SigLIP provide widely used representations for multimodal learning systems. While these models are typically compared through downstream performance, such evaluations often do not explain how their representations differ structurally. In this work, we study this problem through the task of Contrastive Embedding Clustering: identifying sample subsets that are weakly clustered under one representation but strongly clustered under another. We propose \emph{Kernel Optimization for Discrepancy Analysis (KODA)}, a kernel-based framework for contrastive representation comparison and alignment. KODA constructs unified multimodal kernels through modality-wise kernel composition and formulates discrepancy discovery as a constrained optimization problem that searches for coherent structures in one representation while suppressing coherence in a reference representation. This yields interpretable discrepancy directions associated with specific sample subsets and modality interactions. To scale KODA to large vision-language datasets, we develop randomized low-dimensional approximations of joint kernels using random projections, including Random Fourier Features for shift-invariant kernels. Empirically, KODA identifies consistent and interpretable discrepancy structures across vision-language representations and provides sample subsets for representation alignment. The code is available at https://github.com/yokiwuuu/KODA.

We study the problem of differentially private (DP) $k$-means clustering in Euclidean space. Previous solutions rely on summing the private data directly, which induces a sensitivity proportional to the domain. We introduce PE-means, an extension of the private evolution (PE) algorithm (an increasingly popular method for synthetic data generation), to the problem of $k$-means clustering. The key advantage of PE is that it only computes a private histogram with constant sensitivity to guide the evolution. Our adaptation of PE includes new evolutionary operators for clustering, as well as other algorithmic improvements of independent interest. Overall, PE-means achieves an average improvement of 26% in clustering loss over state-of-the-art baselines such as Google's LSH-based algorithm and DP-Lloyd variants.

We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For this subclass we prove a sample-level almost-sure convergence theorem and an explicit almost-sure rate of convergence, under coupled rate conditions on the penalty parameter, the random-feature count, the sample size, and the optimization tolerance. The framework includes the kernel-MMD-penalty Schrödinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schrödinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.

Score-based and flow-based generative models exhibit remarkable expressive capacity in capturing complex distributions, and have been extensively deployed in tasks ranging from image generation to reinforcement learning. Nevertheless, these models suffer from prolonged inference latency, which imposes a significant computational bottleneck in RL with iterative sampling. To overcome this limitation, we propose a new framework named Moment Matching Q-Learning (MoMa QL), which utilizes a technique from statistical hypothesis testing known as maximum mean discrepancy (MMD) that intend to match all orders of statistics between the original and target distribution. By enforcing strong regularization on all moment statistics, this algorithm guarantees distribution-level convergence for conditional score function and remains stable under various hyperparameters. Empirically, we show that our method MoMa QL is more computationally efficient with a comparable if not competitive performance in various D4RL tasks. Remarkably, by accelerating the action sampling process for flow-based policies, MoMa QL demonstrates superior performance in offline-to-online RL tasks because of faster and stronger adaptability for online interactive finetuning.

Pathology foundation models (PFMs) have advanced rapidly in recent years and support training classifiers for a range of histopathology tasks. However, their robustness across hospitals remains limited: performance often degrades when training a classifier on data from one hospital and evaluating it on another target hospital. We address this challenge by fine-tuning PFMs with a local maximum mean discrepancy (LMMD) objective that applies to two settings: domain adaptation, where unlabeled target-hospital data is available, and domain generalization, where target-hospital data is unavailable at all. Experiments at both the patch- and slide-level show consistent improvements across multiple PFMs and tasks.

Kernel Stein discrepancy (KSD) is among the most popular goodness-of-fit (GoF) measures on general domains with a large number of successful deployments. One of the main applications of KSD is in constructing powerful GoF tests. However, tests relying on the classical U-/V-statistic-based KSD estimators have two major drawbacks. (i) Their runtime scales quadratically in the number of samples. (ii) Their asymptotic null distribution is computationally intractable in most cases, typically handled by bootstrapping. While it is known that the Nyström method permits accelerating KSD estimation with no loss of statistical accuracy under mild conditions, to the best of our knowledge, the fundamental question of its impact on bootstrap-based GoF testing is open; resolving this question is the focus of the current paper. In particular, we prove that the key properties of the quadratic-time bootstrapped KSD-based GoF test (asymptotic level and local consistency) are preserved by its Nyström acceleration. We numerically demonstrate the efficiency of the accelerated KSD estimator and bootstrap in the context of GoF testing of spherical and functional data. Our numerical results show that the Nyström-accelerated method performs statistically on-par with the quadratic-time approach, while requiring substantially smaller runtime.

The Gini Multidimensional Scaling (Gini MDS) framework extends the Euclidean multidimensional scaling. We introduce a Gini pseudo-distance based on values and their ranks that depends on a fine-tunable hyperparameter. This pseudo-distance allows flexible exploration of latent configurations, enabling embeddings that best match observed dissimilarities. The Gini MDS is shown to be robust to noise and outliers, making it well-suited for real-world applications. We provide experiments on 16 UCI datasets with outliers and on MNIST images with noise to show that the Gini MDS outperforms the Euclidean MDS on noisy data. Finally, a tensor-based implementation in \texttt{PyTorch} provides GPU acceleration and efficient computation compared to the standard MDS of the \texttt{sklearn} library.

We propose and analyze a conservative drifting method for one-step generative modeling. The method replaces the original displacement-based drifting velocity by a kernel density estimator (KDE)-gradient velocity, namely the difference of the kernel-smoothed data score and the kernel-smoothed model score. This velocity is a gradient field, addressing the non-conservatism issue identified for general displacement-based drifting fields. We prove continuous-time finite-particle convergence bounds for the conservative method on $\R^d$: a joint-entropy identity yields bounds for the empirical Stein drift, the smoothed Fisher discrepancy of the KDE, and the squared center velocity. The main finite-particle correction is a reciprocal-KDE self-interaction term, and we give deterministic and high-probability local-occupancy conditions under which this term is controlled. We keep the quadrature constants explicit and track their possible bandwidth dependence: the root residual-velocity rate $N^{-1/(d+4)}$ holds under an additional $h$-uniform quadrature regularity condition, while a more general growth condition yields the optimized root rate $N^{-(2-β)/(2(d+4-β))}$, where $0\le β<2$. We also analyze the non-conservative drifting method with Laplace kernel, corresponding to the original displacement-based velocity proposed in Deng et al., 2026 (arxiv:2602.04770). For this method, a sharp companion kernel decomposes the velocity into a positive scalar preconditioning of a sharp-score mismatch plus a Laplace scale-mismatch residual, producing an analogous finite-particle rate with an unavoidable residual term. Finally, we explain how the continuous-time residual-velocity bounds translate into one-step generation guarantees through the explicit drift size $η$.

Integration against a probability distribution given its unnormalized density is a central task in Bayesian inference and other fields. We introduce new methods for approximating such expectations with a small set of weighted samples -- i.e., a quadrature rule -- constructed via an interacting particle system that minimizes maximum mean discrepancy (MMD) to the target distribution. These methods extend the classical mean shift algorithm, as well as recent algorithms for optimal quantization of empirical distributions, to the case of continuous distributions. Crucially, our approach creates dynamics for MMD minimization that are invariant to the unknown normalizing constant; they also admit both gradient-free and gradient-informed implementations. The resulting mean shift interacting particle systems converge quickly, capture anisotropy and multi-modality, avoid mode collapse, and scale to high dimensions. We demonstrate their performance on a wide range of benchmark sampling problems, including multi-modal mixtures, Bayesian hierarchical models, PDE-constrained inverse problems, and beyond.

A likelihood-free transport filtering method is proposed based on the couplings between state and observation variables. By exploiting a block-triangular structure in the transport map, the analysis step of filtering is reformulated as the minimization of the maximum mean discrepancy (MMD) between the true joint measure and its transport-based approximation. To circumvent the non-convexity in the MMD optimization, we introduce a training-free transport filter method via gradient flows, which leads to an analytic computation for the transport map that implies the steepest descent direction of the MMD. The proposed approach accurately approximates non-Gaussian filtering posteriors and avoids particle collapse. We provide a convergence analysis for the expectation of the MMD between the approximated posterior and the truth posterior. Finally, we extend the method to high-dimensional problems through domain localization. Numerical examples demonstrate the superior performance of our approach over conventional filtering methods in nonlinear, non-Gaussian scenarios.

Determinantal point processes (DPPs) have emerged as a kernelized alternative to vanilla independent sampling for generating efficient minibatches, coresets and other parsimonious representations of large-scale datasets. While theoretical foundations and promising empirical performance have been demonstrated, there are two challenges for current proposals for DPP-based coresets or minibatches. The first is the need for families of DPPs with certain key variance reduction properties, usually constructed in a continuous setting, of which there are few known examples. The second is the need for an ad-hoc construction of a discrete DPP defined on a given dataset, that inherits such variance reduction. In this work, we contribute to the programme of establishing DPPs as a subsampling toolbox for ML by advancing on these two fronts. First, we propose new DPPs on the Euclidean space based on wavelets, with provably better accuracy guarantees than the best known rates. Second, we introduce a general method to convert such continuous DPPs, which are more amenable to proving analytical statements, into discrete kernels, which are pertinent for subsampling tasks such as minibatch and coreset constructions. This conversion mechanism simultaneously preserves the desired variance decay and reveals a low-rank decomposition of the discrete kernel, which makes sampling the corresponding DPP computationally inexpensive. En route, we enlarge the class of ML tasks amenable to improvements via DPP-based minibatches and coresets to include objective functions with arbitrarily low regularity, and rate guarantees that explicitly adapt to this regularity.

Sampling from constrained distributions has a wide range of applications, including in Bayesian optimization and robotics. Prior work establishes convergence and feasibility guarantees for constrained sampling, but assumes that the feasible set is connected. However, in practice, the feasible set often decomposes into multiple disconnected components, which makes efficient sampling under constraints challenging. In this paper, we propose MAnifold Sampling via Entropy Maximization (MASEM) for sampling on a manifold with an unknown number of disconnected components, implicitly defined by smooth equality and inequality constraints. The presented method uses a resampling scheme to maximize the entropy of the empirical distribution based on k-nearest neighbor density estimation. We show that, in the mean field, MASEM decreases the KL-divergence between the empirical distribution and the maximum-entropy target exponentially in the number of resampling steps. We instantiate MASEM with multiple local samplers and demonstrate its versatility and efficiency on synthetic and robotics-based benchmarks. MASEM enables fast and scalable mixing across a range of constrained sampling problems, improving over alternatives by an order of magnitude in Sinkhorn distance with competitive runtime.

We propose Sobolev-regularized Maximum Mean Discrepancy (SrMMD) gradient flow, a regularized variant of maximum mean discrepancy (MMD) gradient flow based on a gradient penalty on the witness function. The proposed regularization mitigates the non-convexity of the MMD objective and yields provable \emph{global} convergence guarantees in MMD in both continuous and discrete time. A more surprising appeal is that our convergence analysis does not rely on isoperimetric assumptions on the target distribution. Instead, it is based on a regularity condition on the difference between kernel mean embeddings. A key highlight of the proposed flow is that it is applicable in both sampling (from an unnormalized target distribution) -- using Stein kernels -- and generative modeling settings, unlike previous works, where a gradient flow is suitable for only generative modeling or sampling but not both. The effectiveness of the proposed flow is empirically verified on a broad range of tasks in both generative modelling and sampling.

We propose kernel-gradient drifting, a one-step generative modeling framework that replaces the fixed Euclidean displacement direction in drifting models with directions induced by the kernel itself. Standard drifting is attractive because it enables fast, high-quality generation without distilling a large pretrained diffusion model, but its theory is currently understood mainly for Gaussian kernels, where the drift coincides with smoothed score matching and is identifiable. Our gradient-based reformulation exposes this score-based structure for general kernels: the resulting drift is the score difference between kernel-smoothed data and model distributions, yielding identifiability for characteristic kernels and a smoothed-KL descent interpretation of the drifting dynamics. Since kernel gradients are intrinsic tangent vectors, the same construction extends naturally to Riemannian manifolds and to discrete data via the Fisher-Rao geometry of the probability simplex. Across spherical geospatial data, promoter DNA and molecule generation, kernel-gradient drifting enables state-of-the-art one-step generation beyond the Euclidean setting without distillation.

Flow Matching has recently emerged as a popular class of generative models for simulating a target distribution $μ_1$ from samples drawn from a source distribution $μ_0$. This framework relies on a fixed coupling between $μ_0$ and $μ_1$, and on a deterministic or stochastic bridge to define an interpolating process between the two distributions. The time marginals of this process can then be approximately sampled by estimating the transition rates, or more generally the generator, of its Markovian projection. This framework has recently been extended to the case of discrete source and target distributions, under the name Discrete Flow Matching (DFM). However, theoretical guarantees for such models remain scarce. In this paper, we study two DFM models on $\mathbb{Z}_m^d = \{0,\ldots,m-1\}^d$, sampled through time discretization, and derive non-asymptotic associated bounds for both of them. In contrast to previous work, we establish non-asymptotic bounds in Kullback--Leibler divergence for the early-stopped version of the target distribution. We also derive explicit convergence guarantees in total variation distance with respect to the true target distribution. Importantly, these bounds rely only on an approximation error assumption, relaxing standard score assumptions used in earlier works, while also yielding improved dependence on the vocabulary size $m$ and the dimension $d$.

The Maximum Mean Discrepancy (MMD) is a cornerstone statistic for nonparametric two-sample testing, but its test power is dictated entirely by the chosen kernel. Because any fixed kernel inherently fails to distinguish certain distributions, the kernel must be dynamically optimized. However, data-driven optimization violates the foundational i.i.d. assumption, forcing a strict trade-off in existing frameworks. Ratio criteria ignore this dependence, inducing overfitting and variance collapse on rich kernel classes. Conversely, aggregation methods bypass the dependence using finite grids, but this strategy cannot scale to continuous search spaces like deep kernels. To break this dichotomy, we establish data-driven kernel selection as a model selection problem. We propose Complexity-Penalized MMD (CP-MMD), a criterion derived by applying the two-sample uniform concentration inequality of preceding works to the post-optimization MMD problem. The resulting penalty bounds the empirical MMD by the complexity of the kernel search space, mathematically absorbing the cost of optimization, so that CP-MMD enables direct, grid-free maximization over continuous parametric classes, including scalar bandwidths, polynomial feature bandwidths, and deep network parameters. By formally accounting for optimization complexity, we prove that CP-MMD maximizes true test power while ensuring unconditional Type-I validity. Consequently, CP-MMD enables grid-free kernel selection across linear, polynomial-feature, and deep regimes, matching or exceeding state-of-the-art test power.

Estimating individualized treatment effects from longitudinal observational data is central to data-driven medicine, yet existing methods face a fundamental limitation: reducing confounding bias often suppresses clinically informative heterogeneity, degrading patient-specific predictions. Here, we identify this tension as a bias-precision paradox in causal representation learning and introduce sampling-based maximum mean discrepancy (sMMD), a stochastic alignment strategy that replaces global adversarial balancing with subset-level matching. We instantiate this approach in a framework for counterfactual outcome prediction with attribution-grounded interpretability. Across two large-scale ICU cohorts (n = 27,783), our framework improves accuracy under distribution shift, reducing error by up to 11.5% and substantially increasing recall in high-risk tasks. Mechanistic analyses show that sMMD selectively preserves clinically decisive variables. In human-AI evaluation, our method outperforms clinicians-in-training and large language models, and improves clinician accuracy by 14.7% while reducing decision time, enabling interpretable, real-time clinical decision support.

Labeling medical images is a major bottleneck in the field of medical imaging, as it requires domain-specific expertise, and it gets further complicated due to variability across different medical centers and different imaging devices. Such heterogeneity introduces domain shifts and modality discrepancies, which limits the generalization of trained models. To address this important challenge, we propose an unsupervised domain adaptation framework that combines transfer learning with a Reproducing Kernel Hilbert Space based Maximum Mean Discrepancy loss for the alignment of source and target domains. By jointly optimizing classification and RKHS-MMD losses, the methodology enhances generalization to unannotated medical datasets while diminishing reliance on manual annotation. Experimental evaluations presented on two chest X-ray datasets, which are obtained from different medical centers, show outstanding improvements over models trained without adaptation. Furthermore, we perform a comparative study to see that RKHS-MMD performs better than the standard Maximum Mean Discrepancy in reducing modality gap, emphasizing its effectiveness for medical image classification and also its strong capability in advanced AI-driven medical diagnostics.

Effective sample size is a standard summary of Markov chain Monte Carlo output, but it is usually attached to scalar or Euclidean summaries chosen by the analyst. For manifold-valued samples this choice is not canonical: coordinate-wise effective sample sizes can change under rotations, chart changes, or alternative embeddings of the same underlying path. We propose an intrinsic effective sample size based on kernel discrepancy. The proposed quantity is the number of independent draws that would yield the same expected squared kernel discrepancy between the empirical distribution and the target distribution. This gives an exact finite-sample risk interpretation, an asymptotic integrated-autocorrelation representation, and a coordinate-free diagnostic whenever the kernel respects the geometry of the state space. We establish invariance under transported kernels, operator and principal-direction interpretations, and consistency of a lag-window estimator under boundedness and absolute-regularity conditions. We also discuss valid kernel constructions on manifolds, emphasizing that geodesic Gaussian kernels are not generally positive definite on curved spaces. Sphere experiments illustrate rotation invariance and calibration of the proposed diagnostic against empirical distributional error.

Comparing conditional distributions is a fundamental challenge in statistics and machine learning, with applications across a wide range of domains. While proposed methods for measuring discrepancies using kernel embeddings of distributions in a reproducing kernel Hilbert space (RKHS) provide powerful non-parametric techniques, the existing literature remains fragmented and lacks a unified theoretical treatment. This paper addresses this gap by establishing a coherent framework for studying kernel-based methods to measure divergence between conditional distributions through what we refer to as conditional maximum mean discrepancy (CMMD). The CMMD consists of a family of metrics which we call levels, with three special cases each using a different type of RKHS embedding: CMMD$_0$ (conditional mean operators), CMMD$_1$ (conditional mean embeddings), and CMMD$_2$ (joint mean embeddings). We additionally introduce a general level $s$ CMMD, clarifying the required assumptions, and establishing mathematical connections between the levels through the lens of operator-based smoothing. In addition to reviewing previously proposed estimators, we introduce a novel doubly robust estimator for the CMMD that maintains consistency provided at least one of the underlying models is correctly specified. We provide numerical experiments demonstrating that the CMMD effectively captures complex conditional dependencies for statistical testing.

In modern parametric model training, full-batch gradient descent (and its variants) suffers due to progressively stronger biasing towards the exact realization of training data; this drives the systematic ``generalization gap'', where the train error becomes an unreliable proxy for test error. Existing approaches either argue this gap is benign through complex analysis or sacrifice data to a validation set. In contrast, we introduce decoupled descent (DD), a novel theory-based training algorithm that satisfies a train-test identity -- enforcing the train error to asymptotically track the test error for stylized Gaussian mixture models. Within this specific regime, leveraging approximate message passing theory, DD iteratively cancels the biases due to data reuse, rigorously demonstrating the feasibility of zero-cost validation and $100\%$ data utilization. Moreover, DD is governed by a low-dimensional state evolution recursion, rendering the dynamics of the algorithm transparent and tractable. We validate DD on XOR classification, yielding superior performance compared to GD; additionally, we implement noisy MNIST and non-linear probing of CIFAR-10, demonstrating that even when our stylized assumptions are relaxed, DD narrows the generalization gap compared to GD.

Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry. Particularly in the context of machine learning, they are central to clustering, exponential families, parameter estimation and optimisation, among other things. Despite this, the full toolkit of Hilbert spaces and in particular reproducing kernel Hilbert spaces have not been systematically developed and applied to functional Bregman divergences, where points are functions rather than finite-dimensional parameter vectors. While other types of functional Bregman divergences have been studied, these are typically in a Banach space rather than more directly aligned with kernel methods and Hilbert-space geometry commonly used in machine learning. We consider functional Bregman divergences on a Hilbert space, where the self-dual pairing and Riesz representer afford us particularly convenient calculus. Further specialising Bregman generators as a composition involving a kernel mean embedding makes such divergences easy to estimate. We discuss applications in clustering, universal estimation, robust estimation and generative modelling, and contrast our approach with other types of Bregman divergences.

Multivariate conformal prediction requires nonconformity scores that compress residual vectors into scalars while preserving certain implicit geometric structure of the residual distribution. We introduce a Multivariate Kernel Score (MKS) that produces prediction regions that explicitly adapt to this geometry. We show that the proposed score resembles the Gaussian process posterior variance, unifying Bayesian uncertainty quantification with the coverage guarantees of frequentist-type. Moreover, the MKS can be decomposed into an anisotropic Maximum Mean Discrepancy (MMD) that interpolates between kernel density estimation and covariance-weighted distance. We prove finite-sample coverage guarantees and establish convergence rates that depend on the effective rank of the kernel-based covariance operator rather than the ambient dimension, enabling dimension-free adaptation. On regression tasks, the MKS reduces the volume of prediction region significantly, compared to ellipsoidal baselines while maintaining nominal coverage, with larger gains at higher dimensions and tighter coverage levels.

This work focuses on per-video unsupervised action segmentation, which is of interest to applications where storing large datasets is either not possible, or nor permitted. We propose to segment videos by learning in deep kernel space, to approximate the underlying frame distribution, as closely as possible. To define this closeness metric between the original video distribution and its approximation, we rely on maximum mean discrepancy (MMD) which is a geometry-preserving metric in distribution space, and thus gives more reliable estimates. Moreover, unlike the commonly used optimal transport metric, MMD is both easier to optimize, and faster. We choose to use neural tangent kernels (NTKs) to define the kernel space where MMD operates, because of their improved descriptive power as opposed to fixed kernels. And, also, because NTKs sidestep the trivial solution, when jointly learning the inputs (video approximation) and the kernel function. Finally, we show competitive results when compared to state-of-the-art per-video methods, on six standard benchmarks. Additionally, our method has higher F1 scores than prior agglomerative work, when the number of segments is unknown.

\We introduce the horospherical depth, an intrinsic notion of statistical depth on Hadamard manifolds, and define the Busemann median as the set of its maximizers. The construction exploits the fact that the linear functionals appearing in Tukey's half-space depth are themselves limits of renormalized distance functions; on a Hadamard manifold the same limiting procedure produces Busemann functions, whose sublevel sets are horoballs, the intrinsic replacements for halfspaces. The resulting depth is parametrized by the visual boundary, is isometry-equivariant, and requires neither tangent-space linearization nor a chosen base point. For arbitrary Hadamard manifolds, we prove that the depth regions are nested and geodesically convex, that a centerpoint of depth at least $1/(d+1)$ exists, and hence that the Busemann median exists for every Borel probability measure. Under strictly negative sectional curvature and mild regularity assumptions, the depth is strictly quasi-concave and the median is unique. We also establish robustness: the depth is stable under total-variation perturbations, and under contamination escaping to infinity the limiting median depends on the escape direction but not on how far the contaminating mass has moved along the geodesic ray, in contrast with the Fréchet mean. Finally, we establish uniform consistency of the sample depth and convergence of sample depth regions and sample Busemann medians; on symmetric spaces of noncompact type, the argument proceeds through a VC analysis of upper horospherical halfspaces, while on general Hadamard manifolds it follows from a compactness argument under a mild non-atomicity assumption.

We study discrete k-clustering problems in general metric spaces that are constrained by a combination of two different fairness conditions within the demographic fairness model. Given a metric space (P,d), where every point in P is equipped with a protected attribute, and a number k, the goal is to partition P into k clusters with a designated center each, such that a center-based objective function is minimized and the attributes are fairly distributed with respect to the following two fairness concepts: 1) group fairness: We aim for clusters with balanced numbers of attributes by specifying lower and upper bounds for the desired attribute proportions. 2) diverse center selection: Clusters have natural representatives, i.e., their centers. We ask for a balanced set of representatives by specifying the desired number of centers to choose from each attribute. Dickerson, Esmaeili, Morgenstern and Zhang (2023) denote the combination of these two constraints as doubly constrained fair clustering. They present algorithms whose guarantees depend on the best known approximation factors for either of these problems. Currently, this implies an 8-approximation with a small additive violation on the group fairness constraint. For k-center, we improve this approximation factor to 4 with a small additive violation. This guarantee also depends on the currently best algorithm for DS-fair k-center given by Jones, Nguyen and Nguyen (2020). For k-median and k-means, we propose the first constant-factor approximation algorithms. Our algorithms transform a solution that satisfies diverse center selection into a doubly constrained fair clustering using an LP-based approach. Furthermore, our results are generalizable to other center-selection constraints, such as matroid k-clustering and knapsack constraints.

The L infinity star discrepancy is a measure for how uniformly a point set is distributed in a given space. Point sets of low star discrepancy are used as designs of experiments, as initial designs for Bayesian optimization algorithms, for quasi-Monte Carlo integration methods, and many other applications. Recent work has shown that classical constructions such as Sobol', Halton, or Hammersley sequences can be outperformed by large margins when considering point sets of fixed sizes rather than their convergence behavior. These results, highly relevant to the aforementioned applications, raise the question of how much existing constructions can be improved through size-specific optimization. In this work, we study this question for the so-called Kronecker construction. Focusing on the 3-dimensional setting, we show that optimizing the two configurable parameters of its construction yields point sets outperforming the state-of-the-art value for sets of at least 500 points. Using the algorithm configuration technique irace, we then derive parameters that yield new state-of-the-art discrepancy values for whole ranges of set sizes.

Accurately and efficiently estimating the variance of the Maximum Mean Discrepancy (MMD) remains challenging, particularly for unbalanced sample sizes. In this paper, we derive a finite-sample unbiased estimator of the MMD variance. To overcome the traditional $\mathcal{O}(N^2)$ computational bottleneck, we develop a recursive prefix-suffix accumulation scheme for the Laplace kernel, reducing the computational complexity to $\mathcal{O}(N \log N)$ while requiring $\mathcal{O}(N)$ memory. Experimental results verify the theoretical exactness and numerical stability of the proposed estimator and demonstrate its scalability on large datasets. Furthermore, the method proves effective for monitoring distributional convergence during the training of Time-series Generative Adversarial Networks (TimeGAN).

Anticlustering is an NP-hard combinatorial optimization problem that consists of partitioning a set of objects into equal-sized groups called anticlusters such that the objects in the same anticluster are as dissimilar as possible and thereby representative of the entire set of objects. Here we study the case where the dissimilarity metric is the squared Euclidean distance between the respective feature vectors. Applications of Euclidean anticlustering include social studies, cross-validation, creating mini-batches for stochastic gradient descent, and finding balanced K-cut partitions. In particular, machine-learning applications such as mini-batch generation involve million-scale datasets and very large values of K, making scalable anticlustering algorithms essential. We propose a new algorithm, the Assignment-Based Anticlustering (ABA) algorithm, that scales to instances with millions of objects and hundreds of thousands of anticlusters within seconds to minutes, which is far beyond what existing anticlustering methods can manage. We demonstrate here, via an extensive computational study, that our algorithm outperforms existing anticlustering methods in both solution quality and running time. This is so also for anticlustering with categories. For the related problem of balanced K-cut partitioning, our algorithm is superior to the well-known METIS method. The code of our algorithm is available on GitHub.

Existing two-sample testing techniques, particularly those based on choosing a kernel for the Maximum Mean Discrepancy (MMD), often assume equal sample sizes from the two distributions. Applying these methods in practice can require discarding valuable data, unnecessarily reducing test power. We address this long-standing limitation by extending the theory of generalized U-statistics and applying it to the usual MMD estimator, resulting in new characterization of the asymptotic distributions of the MMD estimator with unequal sample sizes (particularly outside the proportional regimes required by previous partial results). This generalization also provides a new criterion for optimizing the power of an MMD test with unequal sample sizes. Our approach preserves all available data, enhancing test accuracy and applicability in realistic settings. Along the way, we give much cleaner characterizations of the variance of MMD estimators, revealing something that might be surprising to those in the area: while zero MMD implies a degenerate estimator, it is sometimes possible to have a degenerate estimator with nonzero MMD as well; we give a construction and a proof that it does not happen in common situations.

Discretizing raw features into bucketized attribute representations is a popular step before sharing a dataset. It is, however, evident that this step can cause significant bias in data and amplify unfairness in downstream tasks. In this paper, we address this issue by introducing the unbiased binning problem that, given an attribute to bucketize, finds its closest discretization to equal-size binning that satisfies group parity across different buckets. Defining a small set of boundary candidates, we prove that unbiased binning must select its boundaries from this set. We then develop an efficient dynamic programming algorithm on top of the boundary candidates to solve the unbiased binning problem. Finding an unbiased binning may sometimes result in a high price of fairness, or it may not even exist, especially when group values follow different distributions. Considering that a small bias in the group ratios may be tolerable in such settings, we introduce the epsilon-biased binning problem that bounds the group disparities across buckets to a small value epsilon. We first develop a dynamic programming solution, DP, that finds the optimal binning in quadratic time. The DP algorithm, while polynomial, does not scale to very large settings. Therefore, we propose a practically scalable algorithm, based on local search (LS), for epsilon-biased binning. The key component of the LS algorithm is a divide-and-conquer (D&C) algorithm that finds a near-optimal solution for the problem in near-linear time. We prove that D&C finds a valid solution for the problem unless none exists. The LS algorithm then initiates a local search, using the D&C solution as the upper bound, to find the optimal solution.

Clustering algorithms often assume all features contribute equally to the data structure, an assumption that usually fails in high-dimensional or noisy settings. Feature weighting methods can address this, but most require additional parameter tuning. We propose SHARK (Shapley Reweighted $k$-means), a feature-weighted clustering algorithm motivated by the use of Shapley values from cooperative game theory to quantify feature relevance, which requires no additional parameters beyond those in $k$-means. We prove that the $k$-means objective can be decomposed into a sum of per-feature Shapley values, providing an axiomatic foundation for unsupervised feature relevance and reducing Shapley computation from exponential to polynomial time. SHARK iteratively re-weights features by the inverse of their Shapley contribution, emphasising informative dimensions and down-weighting irrelevant ones, and is equivalent to replacing the arithmetic mean of feature dispersions with their harmonic mean. Experiments on synthetic and real-world data sets show that SHARK consistently matches or outperforms existing methods, achieving superior robustness and accuracy, particularly in scenarios where noise may be present. Software: https://github.com/rickfawley/SHARK.

We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernelized Stein Discrepancy ($\mathsf{KSD}$) and Wasserstein-2 metrics. Our key insight is that the time derivative of the relative entropy between the joint density of $N$ particle locations and the $N$-fold product target measure, starting from a regular initial distribution, splits into a dominant negative part' proportional to $N$ times the expected $\mathsf{KSD}^2$ and a smaller positive part'. This observation leads to $\mathsf{KSD}$ rates of order $1/\sqrt{N}$, in both continuous and discrete time, providing a near optimal (in the sense of matching the corresponding i.i.d. rates) double exponential improvement over the recent result by Shi and Mackey (2024). Under mild assumptions on the kernel and potential, these bounds also grow polynomially in the dimension $d$. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence in continuous time. For the case of `bilinear + Matérn' kernels, we derive Wasserstein-2 rates that exhibit a curse-of-dimensionality similar to the i.i.d. setting. We also obtain marginal convergence and long-time propagation of chaos results for the time-averaged particle laws.

Clustering problems such as $k$-means and $k$-median are staples of unsupervised learning, and many algorithmic techniques have been developed to tackle their numerous aspects. In this paper, we focus on the class of greedy approximation algorithm, that attracted less attention than local-search or primal-dual counterparts. In particular, we study the recursive greedy algorithm developed by Mettu and Plaxton [SIAM J. Comp 2003]. We provide a simplification of the algorithm, allowing for faster implementation, in graph metrics or in Euclidean space, where our algorithm matches or improves the state-of-the-art.

Providing various machine learning (ML) applications in the real world, concerns about discrimination hidden in ML models are growing, particularly in high-stakes domains. Existing techniques for assessing the discrimination level of ML models include commonly used group and individual fairness measures. However, these two types of fairness measures are usually hard to be compatible, and even two different group fairness measures might be incompatible as well. To address this issue, we investigate and evaluate the discrimination level of classifiers from a manifold perspective and propose a fairness measure named harmonic fairness via manifolds (HFM)'' based on distances between sets. Yet the direct calculation of distances might be too expensive to afford, reducing its practical applicability. Therefore, we devise an approximation algorithm named Approximation of distance between sets (ApproxDist)'' to facilitate accurate estimation of distances, and we further demonstrate its algorithmic effectiveness under certain reasonable assumptions. Empirical results indicate that the proposed fairness measure HFM reflects bias from both individual and group fairness aspects and that the proposed ApproxDist is effective and efficient.

Kernel herding belongs to a family of deterministic quadratures that seek to minimize the maximum mean discrepancy (MMD), that is, the worst-case integration error over a reproducing kernel Hilbert space (RKHS). These MMD minimization procedures come with strong experimental support, but comparatively less theoretical footing. In particular, apart from recent progress in distribution compression, little has been proved in favor of an improvement of MMD minimization over classical Monte Carlo quadrature when the RKHS is infinite-dimensional. In this paper, we study a joint probability distribution over quadrature nodes, a tailored Gibbs distribution, whose support intuitively tends to concentrate around MMD minimizers as a temperature parameter is decreased. Our main contribution is to prove that drawing integration nodes from our distribution does outperform i.i.d Monte Carlo. While our bounds on the worst-case integration error feature the same rate as i.i.d. Monte Carlo, we do obtain a tighter concentration inequality as the temperature parameter decreases. This means smaller confidence intervals as the number of quadrature nodes increases. While arguably a first step, our results demonstrate that the mathematical toolbox developed around Gibbs measures can help understand to what extent kernel herding and its variants improve on computationally cheaper methods. There remains the issue of sampling from our Gibbs distribution. In our numerical experiments, we demonstrate that a simple MCMC chain already yields approximate samples that lead to improved confidence intervals around the target integrals, as supported by our theoretical results.

In recent years, there has been a surge in effort to formalize notions of fairness in machine learning. We focus on centroid clustering--one of the fundamental tasks in unsupervised machine learning. We propose a new axiom ``proportionally representative fairness'' (PRF) that is designed for clustering problems where the selection of centroids reflects the distribution of data points and how tightly they are clustered together. Our fairness concept is not satisfied by existing fair clustering algorithms. We design efficient algorithms to achieve PRF both for unconstrained and discrete clustering problems. Our algorithm for the unconstrained setting is also the first known polynomial-time approximation algorithm for the well-studied Proportional Fairness (PF) axiom. Our algorithm for the discrete setting also matches the best known approximation factor for PF.